Search Results (327)

The main aim of this study is to create appropriate criteria for the existence of a unique $\mu$-pseudo almost periodic solution to a particular type of stochastic differential equation, utilizing Bochner’s double sequences criterion, improved Gronwall’s lemma, Hölder’s inequality, and measure theory techniques. By applying the inequalities analysis condition and the fixed point theorem for contraction mapping, we can establish the existence of a single $\mu$-pseudo almost periodic solution in distribution to the given stochastic equation. Finally, we use an example to demonstrate our stochastic processes. Full article

In this study, we investigate implicit fractional differential equations subject to anti-periodic boundary conditions. The fractional operator incorporates two distinct generalizations: the Caputo tempered fractional derivative and the Caputo fractional derivative with respect to a smooth function. We investigate the existence and uniqueness of solutions using fixed-point theorems. Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias is also considered. Three detailed examples are presented to illustrate the applicability and scope of the theoretical results. Several existing results in the literature can be recovered as particular cases of the framework developed in this work. Full article

The main objective of this work is to discuss the generalized anti-periodic boundary conditions of the generalized Caputo fractional differential equations with $p\left(t\right)$-Laplacian operators. By applying the Schaefer fixed point theorem, the existence of mild solutions to this problem is obtained, which generalizes and enriches the anti-periodic boundary value problem of Caputo fractional hybrid differential equations. Finally, a numerical example is given to verify our main results. The anti-periodic boundary value condition imparts a form of symmetric inversion with respect to the original state, so it exhibits an anti-symmetric structural feature. Full article

In LEO constellation–augmented navigation, the transmitting antenna phase center offset (PCO) and the equipment delay associated with the downlink signals of LEO satellites constitute major error sources that must be precisely characterized. Previous studies primarily focused on single or small-scale satellite scenarios, lacking comprehensive evaluations regarding the influence of constellation scale, orbital altitude, ground station configuration, and various error sources. To address this gap, we propose a joint estimation method utilizing observations from a limited number of regional ground stations in China that simultaneously track GNSS and LEO satellites. The method is specifically designed to accommodate practical constraints on ground station distribution within China. Initially, a batch least-squares estimation strategy is employed to simultaneously determine the ionosphere-free PCO and initial equipment delays for all LEO satellites in a constellation-wide solution. Subsequently, the estimated PCO parameters are fixed, and the equipment delays are further refined using a precise point positioning (PPP) approach. To systematically evaluate the method’s performance under realistic conditions, we analyze the impact of orbital altitude, constellation size, ground station number, data processing duration, and orbit/clock biases through comprehensive simulations. The results indicate: (1) the Z-direction component of the PCO (pointing toward the Earth’s center) and equipment delay is more sensitive to orbit and clock errors; (2) Increasing the number of LEO satellites generally improves the estimation accuracy of equipment delays, but the marginal gain diminishes as the constellation size expands; (3) sub-centimeter PCO accuracy and equipment delay accuracies better than 3 cm can still be achieved using only 3–4 regionally distributed ground stations over an observation period of 5–7 days. Full article

Introduction. Pediatric polytrauma (PPT) and major trauma in pediatric patients (PMT) present unique challenges compared to adult trauma care due to distinct anatomical and physiological differences. PPT/PMT remains the leading cause of death in children, responsible for over 50% of pediatric deaths and 15% of pediatric hospital admissions due to its long-term effects. This single-institution study focuses on the initial management of PPT/PMT from an orthopedics and traumatology point of view. Material and Methods. In the present study, data of PPT/PMT managed in one single institution, an academic level I pediatric trauma center, in patients <18 years of age, were analyzed over different periods. Over a 10-year period, diaphyseal femur fractures were analyzed as indicators of damage control (DCO) versus definitive treatment. Over a 4-year period (2021–2024), the associated lesions of PPT (head injuries, thoracic and abdominal lesions, spine lesions, major blood vessel lesions, and major musculoskeletal injury) were analyzed. Over a 1-year period (2019), the overall in-hospital mortality and admission rates in the pediatric intensive care unit were analyzed. Results. In the 10-year period, among 298 diaphyseal femur fractures, 46/298 (15%) were classified as PPT in which DCO was performed according to age as follows: in the age-group 15–17 years 23/23 (100%) with temporary external fixation (ExFix); in the age group 12–14 years, 9/14 (64%) with ExFix and 5/14 (26%) and elastic stable intramedullary nails (ESINs); in the age group 5–11 years, 1/5 (20%) with ExFix and 4 with ESIN; in the age group 0–4 years, 2/4 (50%) with ESIN and 2/4 (50%) with a cast. In the 4-year period, PPT/PMTs were associated with 60% head injury, 25% thoracic lesion(s), 18% abdominal lesion(s), 16% spine injury, 5% lesion of a major blood vessel, and 30% major musculoskeletal injuries. In 2019, there were 193 patients admitted to the emergency room as PPT/PMT: 115 were ≤12 years old and 78 were >12 years old. On admission, 46% were admitted to the pediatric intensive care unit, and 65% were admitted to the department of traumatology as inpatients. The in-hospital mortality rate was 7%. Discussion and Conclusions. In our institution, pediatric trauma is assessed using the Pediatric Trauma Score (PTS), and the workup follows the ATLS guidelines with a dedicated trauma team. The role of the orthopedic surgeon during the primary evaluation of PPT/PMT is to contribute to stopping bleeding and hemorrhagic shock. In PPT/PMT, DCO in adolescents is superimposable to adults, whilst in babies and children, DCO is still performed, but it is not a form of temporary external fixation. Full article

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order $\alpha \in (0,1)$ and $\alpha \in (1,2)$, both defined with respect to a general kernel function. The existence and uniqueness of solutions are established using Banach’s and Schaefer’s fixed-point theorems under suitable Lipschitz conditions. Furthermore, Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated for each problem. Examples are provided to illustrate the applicability of the main results. Full article

Differential equations have demonstrated significant practical effectiveness across diverse fields, including physics, chemistry, biological engineering, computer science, electrical power systems, and security cryptography. This study investigates the dynamics of a Caputo fractional discrete-time modified Brusselator model. Conditions for the existence and local stability of the fixed point $FP$ are established. Using bifurcation theory, the occurrence of both period-doubling and Neimark–-Sacker bifurcations is analyzed, revealing that these bifurcations occur at specific values of the bifurcation parameter. Maximum Lyapunov characteristic exponents are computed to assess system sensitivity. Two-dimensional diagrams are presented to illustrate phase portraits, local stability regions, closed invariant curves, bifurcation types, and Lyapunov exponents, while the 0-1 test confirms the presence of chaos in the model. Finally, MATLAB simulations validate the theoretical results. Full article

The 4-body equations of motion are derived in our previously published paper. Here we prove the existence–uniqueness of a periodic solution by applying the fixed-point method for a suitable introduced operator. To apply the fixed-point theorem, we need to derive appropriate analytical inequalities for the right-hand sides of the equations that ensure that the operator for periodic solutions maps the set of periodic functions into itself. In this way, we prove the existence of the Bohr–Sommerfeld orbits for the 4-body problem in the relativistic case. That allows us to estimate the minimal distances between the electrons on the first and second Bohr–Sommerfeld stationary states. A natural example of such a problem is the Lithium atom, which has three electrons orbiting the nucleus. Full article

To quantitatively evaluate the hydrological regime dynamics in West Dongting Lake over the past seven decades, this study utilizes daily average water level series (1955–2024) from key control stations (Nanzui and Xiaohezui) to analyze variations in water level and discharge through change-point detection methods, adopting the water level difference between Xiaohezui and Nanzui as a pivotal indicator of hydrological changes; the IHA–RVA framework is then applied to comprehensively assess the degree of alteration in hydrological indicators before and after identifying change points, demonstrating the following: (1) declining trends in water level/discharge at both stations—primarily attributable to reduced inflows from the Songzi and Hudu Rivers—underwent abrupt shifts in 1983 and 2003, while the water level difference displayed an increasing trend with a change point in 1991; (2) the overall degree of hydrologic alteration (DHA) was moderate, with enhanced variability during T2 (2003–2024) relative to T1 (1983–2003), notably for discharge at Nanzui and water level at Xiaohezui; (3) reduced discharge in the Songzi and Hudu Rivers primarily drives the decreased outflow from West Dongting Lake. In the Li and Yuan basins during period T1, anthropogenic factors dominated runoff alterations. During T2, anthropogenic contributions accounted for 76.27% and 48.67% of runoff changes, respectively, resulting in reduced runoff volumes under equivalent precipitation inputs. (4) Under fixed water level differences, a significant positive correlation exists between discharges at Xiaohezui and Nanzui stations. Greater discharge flows downstream through the flow channel adjacent to NZ at West Dongting Lake’s outlet. Collectively, these findings establish a technical foundation for assessing the impact of hydrological regimes and aquatic ecological security in Dongting Lake, thereby advancing sustainable water resource utilization across the basin. Full article

This paper examines the changing role of emissions trading systems (ETSs) within the macro-financial framework of energy markets, emphasizing price dynamics and systemic spillovers. Utilizing monthly data from seven ETS jurisdictions spanning January 2021 to December 2024 (N = 287 observations after log transformation and first differencing), which includes four auction-based markets (United States, Canada, United Kingdom, South Korea), two secondary markets (China, New Zealand), and a government-set fixed-price scheme (Germany), this research estimates a panel vector autoregression (PVAR) employing a Common Correlated Effects (CCE) model and augments it with machine learning analysis utilizing XGBoost and explainable AI methodologies. The PVAR-CEE reveals numerous unexpected findings related to carbon markets: ETS returns exhibit persistence with an autoregressive coefficient of −0.137 after a four-month lag, while increasing inflation results in rising ETS after the same period. Furthermore, ETSs generate spillover effects in the real economy, as elevated ETSs today forecast a 0.125-point reduction in unemployment one month later and a 0.0173 increase in inflation after two months. Impulse response analysis indicates that exogenous shocks, including Brent oil prices, policy uncertainty, and financial volatility, are swiftly assimilated by ETS pricing, with effects dissipating completely within three to eight months. XGBoost models ascertain that policy uncertainty and Brent oil prices are the most significant predictors of one-month-ahead ETSs, whereas ESG factors are relevant only beyond certain thresholds and in conditions of low policy uncertainty. These findings establish ETS markets as dynamic transmitters of macroeconomic signals, influencing energy management, labor changes, and sustainable finance under carbon pricing frameworks. Full article

The aim of this work is to study a class of nabla fractional difference equations with anti-periodic conditions. First, we construct the related Green’s function. After deducing some of its useful properties, we obtain an upper bound for its sum. Then, using this bound, we are able to obtain three existence results based on the Banach contraction principle, Brouwer’s fixed point theorem, and Leray–Schauder’s nonlinear alternative, respectively. Then, we show some non-existence results for the studied problem, and existence results are also provided for a system of two equations of the considered type. Finally, we outline some particular examples in order to demonstrate the theoretical findings. Full article

This paper proposes a Hybrid Recursive Trigonometric (HRT) technique for FPGA-based direct digital frequency synthesizers. The HRT technique integrates a recursive cosine generator with periodic reinitialization via a second-order Taylor polynomial to reduce cumulative errors without requiring ROMs or iterative CORDIC units. A resource-efficient combinational architecture is implemented and validated on the Lattice iCE40HX1K FPGA. The effectiveness of the proposed HRT technique is evaluated through simulation and FPGA-based experiments, with respect to spectral accuracy and resource efficiency, particularly for fixed-point cosine waveform synthesis in low-resource digital systems. Simulation results show that the system has a spurious-free dynamic range (SFDR) of −86.09 dBc and signal-to-noise ratio of 52.74 dB using 16-bit fixed-point arithmetic. Experimental measurements confirm the feasibility, achieving −58.86 dBc SFDR. Full article

Precipitation, as a key input in the water cycle, directly influences the formation and change process of runoff. Meanwhile, the return runoff intuitively reflects the available quantity of water resources in a river basin. An in-depth analysis of the evolution laws and response relationships between precipitation and return runoff over a long time scale serves as an important support for exploring the evolution of hydrometeorological conditions and provides an accurate basis for the scientific planning and management of water resources. Taking Lanzhou Station on the upper Yellow River as a typical case, this study proposes the VSSL (LSTM Fusion Method Optimized by SSA with VMD Decomposition) deep learning precipitation element series extension method and the SSVR (SVR Fusion Method Optimized by SSA) machine learning runoff element series extension method. These methods achieve a reasonable extension of the missing data and construct 100-year precipitation and return runoff series from 1921 to 2020. The research results showed that the performance of machine learning and deep learning methods in the precipitation and return runoff test sets is better than that of traditional statistical methods, and the fitting effect of return runoff is better than that of precipitation. The 100-year precipitation and return runoff series of Lanzhou Station from 1921 to 2020 show a non-significant upward trend at a rate of 0.26 mm/a and 0.42 × 10⁸ m³/a, respectively. There is no significant mutation point in precipitation, while the mutation point of return runoff occurred in 1991. The 100-year precipitation series of Lanzhou Station has four time-scale alternations of dry and wet periods, with main periods of 60 years, 20 years, 12 years, and 6 years, respectively. The 100-year return runoff series has three time-scale alternations of dry and wet periods, with main periods of 60 years, 34 years, and 26 years, respectively. During the period from 1940 to 2000, an approximately 50-year cycle, precipitation and runoff not only have strong common-change energy and significant interaction, but also have a fixed phase difference. Precipitation changes precede runoff, and runoff responds after a fixed time interval. Full article

Clarifying the coordination and impediments of social, economic, and spatial connection in rural areas is essential for advancing rural revitalization, urban-rural integration, and regional coordinated development. Utilizing the 24 counties and districts in western Hunan as case studies, we developed an evaluation index system for sustainable rural development across three dimensions: social, economic, and spatial. We employed the coupling model, coordination model, and obstacle factor model to investigate the comprehensive development level, coupling and coordination status, and obstacle factors of the villages in the study area at three temporal points: 2002, 2012, and 2022. The findings indicate the following: (1) The degree of rural development in western Hunan has escalated swiftly throughout the study period, transitioning from relative homogeneity to a heterogeneous developmental landscape, accompanied by issues such as inadequate development and regional polarization. (2) The overall rural social, economic, and spatial indices are low, and the degree of coupling has increased variably across different study periods; the average coordination degree has gradually improved over time, yet the level of coordination remains low, and spatial development is unbalanced. (3) The criterion-level impediments hindering the sustainable development of rural society, economy, and space are, in descending order, social factors, spatial factors, and economic factors. The urbanization rate, total fixed investment rate, and arable land change rate are the primary impediments in most counties and cities. The study’s findings will inform the planning of rural development in ethnic regions, promote sustainable social and spatial advancement in the countryside, and serve as a reference for rural revitalization efforts. Full article

Non-unique fixed-point theorems play a pivotal role in the mathematical modeling to solve certain typical equations, which admit more than one solution. In such situations, traditional outcomes fail due to uniqueness of fixed points. The primary aim of the present article is to investigate a non-unique fixed-point theorem in the framework of a metric space endowed with a local class of transitive binary relations. To obtain our main objective, we introduce a new nonlinear contraction-inequality that subsumes the ideas involved in four noted contraction conditions, namely: almost contraction, Boyd–Wong contraction, Pant contraction and relational contraction. We also establish the corresponding uniqueness theorem for the proposed contraction under some additional hypotheses. Several examples are furnished to illustrate the legitimacy of our newly proved results. In particular, we deduce a fixed-point theorem for almost Boyd–Wong contractions in the setting of abstract metric space. Our results generalize, enhance, expand, consolidate and develop a number of known results existing in the literature. The practical relevance of the theoretical findings is demonstrated by applying to study the existence and uniqueness of solution of a specific periodic boundary value problem. Full article